Stability of a Class of Dynamic Routing Protocols (IGRP) * Steven Low
Pravin Varaiya
AT&T Bell Labs 600 Mountain Ave Murray Hill, NJ 07974
EECS Department University of California Berkeley, CA 94720 varai y
[email protected]
[email protected] Abstract W e perform an exact analysis of the dynamic behavior of IGRP, an adaptive shortest-path routing algorithm widely used in the industry, on a simple ring network. The distance metric is a weighted sum of traficsensitive and trafic-insensitive delay components. We relate the optimality and stability of the protocol t o the ratio of the weights. In particular, we show that if the trafic-insensitive component is not given enough weight, then starting from any initial routing, the subsequent routings aflerfinitely many update periods will oscillate between two worst cases. Otherwise, the successive routings will converge to the unique equilibrium routing. We also show that load sharing among routes whose distances are within a threshold of the min.inium distance help stabilize the dynamic behavior.
1
Introduction
A packet-switched network transports messages, packaged into streams of packets, between end-points. An end-point may represent a computer, a local area network, an audio or video source, or a database, etc. To each source-destination pair is usually associated a set of routes, or paths across the network. A routing strategy assigns to each packet a route from the set. Each route is assigned a cost called ‘distance’, which is usually a measure of hop-count or delay on the route. Under the shortest path routing, a packet is assigned the route with the least distance. Alternatively, load may be shared among multiple routes with similar distances. In static routing the decision is independent of *Researchsupported by Pacific Gas and Electric Company. The authors are grateful to Shau-Ming Lun, Felix Wu, and Ning Xiao of U.C. Berkeley and Steve Callahan and Omid Razavi of PG&E.
the traffic condition in the network. In dynamic routing the decision adapts to changing traffic condition and can potentially better balance the carried load. Typically the distance metric is defined to be a function of traffic on the route and route distances vary as the traffic condition fluctuates. Time is divided into update periods. At the end of each period, a new routing is computed using route distances in the current period, and then used in the next period to route packets for all source-destination pairs. The next routing thus depends on the current period’s routing. Within a period packets between each source-destination pair follow a fixed route (except possibly for load sharing). Like any delayed feedback system it may be plagued by severe oscillation if not properly designed. Each node z stores a routing table, with an entry ( z , w ) for each destination node z # z. The entry ( z , w ) means that a pa.cket arriving a t node 2 that is destined for node z will be sent to z’s neighbor node w . The routing y(n) in period n, n = O,l,.. ., may represent the routing tables at all nodes. We are interested in the ‘stability’ of y(n) for a class of dynamic routing protocols. The protocol we consider is modeled after IGRP [3] (Inter Gateway Routing Protocol), an adaptive routing algorithm developed by Cisco and used in many of its products. IGRP uses a variant of the updating mechanism in the standard Bellman-Ford a l g e rithm [l, pp. 318-3221. The route distance that is exchanged periodically among neighboring nodes is changing rather than fixed. A simplified description is as follows. Periodically (every 90sec) each node z exchanges with its neighbors the distance D ( z ,z ) between itself and all other nodes z # z. D ( z , z ) represents the shortest distance between node and node i . Based on this information, each node z computes its shortest route to every other node z. It then broadcasts these new distances t o its neighbors in the next update period, and the cycle repeats. Without load sharing the route between each source-destination pair
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is unique - one with the shortest distance. With load sharing the traffic between each source-destination pair may be distributed among more than one route. The distance metric used i n IGRP takes the form
0 1
131 D ( z ,2 ) =
kidi(Z,z
)
+ k,d,(z,z )
where k ; , k , 2 0 are user-settable protocol parameters. The ‘topological delay’ d ; ( x ,z ) is a t,rafficinsensitive delay component that generalizes the notion of hop-count. Each link ( k , m ) between nodes IC and m is assumed to have a fixed transmission capacity p ~ ( k ,2~ )0 in bits per second; ~ ( n . , , ~=) 0 if there is no direct link between the nodes. A route p = ( ( 1 1 , l 2 ) , (Iz, l3), . . . , (In-1, In)} is a sequence of links. The ‘topological delay’ di(z, z ) along route p connecting nodes 2: and z is the t,otal t.ransmission time end-to-end for one bit of data. It is directly proportional to hop-count when each link has the same transmission capacity. Note, however, that the protocol is designed for networks in which propagation delay is negligible compared to transmission delay. d s ( z ,z ) is a traffic-sensitive delay component which measures spare capacity of the r0ut.e under the current routing (see (1-2) in $2). Assuming that the traffic is stationary, we are interested in the dynamic behavior of IGRP. Specifically, we investigate the effect of the protocol paramet,ers k , , ki on the ‘stability’ of y(n), the routing in period n, and the optimality of the equilibrium routing when y(n) is ‘stable’. Our analysis provides insight in setting the protocol parameters. This problem w a s considered in [a]. The protocol there, however, is modeled after one used in the ARPANET and has a different distance metric from IGRP. We will compare our analysis and that in (21 after we have introduced our protocol model in $2. Following [a] we restrict attention to a simple ring network. We believe the intuition obtained from this analysis applies to more general topologies. The paper is organized as follows. A model of IGRP is given in $2. $3 establishes the existence and optimality of the unique equilibrium routing. $4 investiga.tes the dynamic behavior of the algorithm wit,hout load sharing. It is proved under certain condit,ions that if the traffic-insensitive component is not given enough weight, i.e. k , / k i too large, then starting from any initial routing, the subsequent routings after finitely many update periods will oscillate between two worst cases. On the other hand, if the traffic-insensitive
E,,, yrl,
‘We have ignored a factor in the distance met,ric that measures the ‘reliability’of a route.
Y Figure 1: A ring network component is given sufficient weight, i.e. k , / L i SURciently small, then regardless of the initial routing, the successive routings will converge to the unique equilibrium routing. IGRP also allows load sharing among routes whose distances are within a threshold of the shortest distance. We model this feature in $5 and show that load sharing has a stabilizing effect on the dynamic behavior of IGRP. Proofs of all our results can be found in [5] and [4].
2
Protocol Model
The network consists of an undirected ring. For analytical simplicity we consider a continuum of nodes 011 the ring, represented by t E [0,1]; see Figure 1. We assume node 1 (or equivalently node 0) is the only destination and every node i E ( 0 , l ) has a source rate of ~ ( ti n) bits per second. A routing y E [0,1] takes the following simple form: under routing y, a node t < y routes its traffic in the negative, or clockwise, direction and a node t 2 y routes its traffic in the positive, or counter-clockwise, direction. Hence the routing decision at each node is simply to decide whether to send its traffic to the left or to the right neighbor. We assume time is slotted into update periods and all nodes operate synchronously. At the end of each update period, the nodes exchange update information, and compute their new routes for the next period. The link capacity at node t in each direction is ~ ( t ) .
5c.4.2 611
3
The topological delay at t is dJt)
For the rest of this paper, we make the simplifying assumption that the transmission capacity is equal a t all nodes, i.e. p ( t ) E 1. We further assume for stability that r ( s ) d s < 1. The first assumption reduces equations (1-4) to
= JolP-yS)dS
in the negative direction (to node 0), and
Jt
in the positive direction (to node 1). At the end of an update period with routing y, each node t computes the flow at t in the negative and the positive directions
1- ( t ,Y) = f+(tlY) =
Optimality of Equilibrium Routing
lY
where
r(s)ds l[t < Yl
1
r(s)ds l[t
> Yl
The traffic-sensitive delay components, given by the reciprocal of the spare capacity of the route at 1, are
in the two directions. The shortest distance froin t to the destination in the negative and the positive directions are respectively
+
D - ( t , Y) = k.sd,(t, Y) L i d f ( l ) D+(t,y) = ksd,+(ti Y) + kidi+(t) for some IC,, to
ICi
(3) (4)
Give11 the currelit routing y, the new routing p is the solutioll to (5). A routing y* is an equilibrium routing if D--(Y*,Y*)= D+(Y*,Y*). Proposition 1 There exists a unique equilibrium
2 0. The new routing jj is the solution
rozl'ing.
D--(i,Y) = D+(Y,Y).
We next consider the optimality of the equilibrium routing. We will use the expression for the delay (sojourn time) through an M/M/l queue with arrival rate R and service rate /I given by
(5)
[2] considered the same ring network and analyzed the stability of an adpative routing protocol used in the ARPANET. The metric D - ( t , y) ( D + ( t ,y)) used there was the sum, or integral, over all nodes t of a given function d(f-(t, y)) ( d ( f + ( t ,y))) of flow f - ( t , y) (f+(t,y)) at t in the negative (positive) direction. It was proved there that if d(0) = 0, then the routing oscillates between two worst routes [2, Proposition 11, and that if d(0) exceeds certain threshold, then the routing is stable [2, Propositions 5 and 61. The metric used in IGRP is the sum of traffic-sensitive and traffic-insensitive components, weighted by the usersettable protocol parameters IC,, l i . We show that the stability and optimality of the protocol are related to the ratio k , / k i . The same tradeoff between stability and responsiveness to traffic conditions manifest itself here as it did in [2].
1 P-R to interpret the protocol. Under routing y, the delay at node t < y is 1/[1 - f - ( t , y)] in the negative direction. This delay is maximum a t the bottleneck node O+ and is equal to
Similarly, the delay in the positive direction is maximum at the bottleneck node 1- and is equal to
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w +=
1 1 - (R(1) + R(Y))
for nodes 1 E [y, 1). Suppose, however, that each node t E (0, y) knows only its own preferred route (i.e. negative direction) and hence tlie preferred routes of iiodes s < t , but does not know the value of current routing y and hence does not know the preferred route of nodes s > t . Then it would anticipate the delay at the bottleneck node O+ to be at least
where m(y) =
-
J,'W - d t + /y' W + d t Y
1 - R(y)
+
1-Y 1 - R(1) R(Y)
+
It can be shown that solutions of (8) and of (9) coincide if and only if the minimizer y* for (8) also satisfies
1
w-(t) = A 1 - R(t) t E (0,Y)
Similarly, a node t E [y, l ) , knowing only its own preferred route but not the value of y, would anticipate the delay at the bottleneck node 1- to be at. least
W + ( t )=
1
1 - (R(1) - R ( t ) ) '
tE
[Y, 1).
is the ratio of the spare capacity at the bottleneck nodes O+ and 1-.
Hence the total delay at the bottlenecks is at least
=
I'& 1' +
4 dt
1 - R( 1 )
+ H(1)
It is reasonable t o choose a routing y to niini~nize W(y). Since y = f is a good routing when traffic is uniform on the ring or when topological distance is the only cost (k, = 0), we consider the following more general optimization:
where a 2 0 is a given constant. a measures the relative weight we place on the two cost components U'(y) and (y - 1/2)'. According to the following proposition it is related to the relative weight of the two delay components in (6) and (7): a larger a in the object,ive function (8) corresponds to a heavier weight on the traffic-insensitive components in the distance met.ric.
Theorem 2 Let y*(k,, k,) denote t h e unique equilibrium routing with protocol param,elers k , , ki. Then, y*(k,, k,) is the unique minimizer for (8) for e w r y k,, k; satisfying
Stability Without Load Sharing
Recall that a new route is selected every update period by solving ( 5 ) . Proposition 3 Gwen any routang y E [0,1], a new roulziig y E [0, 11 gaven b y the solutzon of (5) exzsts a n d as unzque i f and only af
If t,he condition in Proposition 3 is not satisfied, i.e. I;,
->I;*
1 - R(1) R(1)
then ( 5 ) has no solution y in [0,1] if either D-(O, y) > D+(O,y) or D - ( l , y ) < D t ( l , y ) . We naturally ext,entl tlie new routing t o be $ := 0 in the former case and y := 1 in the latter case. Then starting with any routing y such that 0 - ( 0 , y ) > D+(O,y) or D-( 1, y ) < Dt(1, y), the subsequent routings will oscillate between 0 and 1, the worst possible scenario. I n fact we can give a more complete characterization of the dynamic behavior of the protocol. For y E [O,1],let
Undef routing y the delay at the bottleneck O+ is in fact W - for nodes 15 (0, y), and the delay at the bottleneck 1- is in fact W + for nodes t E [y, 1). I l e n c ~ a better optimization is
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Since both (yzn-lln 2 1) and (yzn,n 2 1) are monotone and bounded, the limits
y := limyZn-1 n
z*= 1
exist and
and
y := l\myZn -
2 5 y* 5 jj. Moreover,
Z l
f(~2n-1) = z~n-2= g(~2n-z) f ( ~ 2 n ) = ZZn-1 = g(y2n-1)
0
and hence the continuity o f f and g implies that ZI
f@) = f(y) =
-1
g(y) g(3-j)
This means that starting with initial routing y1 = p or y1 = y, the subsequent routings will oscillate between and-y. Theorem 5 Suppose tial routing.
Figure 2: Construction of yi and zi
1.
Then (5) is equivalent to
f (Y) = !7(Y)
If y1 = y or y1 = jj, then subsequent routings oscillate between y and B. or y1 > 3-j, then subsequent routings after finitely many update periods oscillate between 0 and 1.
(11)
2. Ifyl
1 there are y E [0,1] such that f(y) > = do)Or f(y) < -1 = d l ) for which no solution y to (11) exists in [0,1]. We define the new routing to be y := 0 in the former case and y := 1 in the latter case. The unique equilibrium routing y* in Proposition 1 satisfies f(y*) = g(y*). Suppose f(1) > g ( 0 ) . Define the sequences (yn, 11 2 1) and (zn, n 2 0) by ZO
2 > wl.Let y1 be an ini-
3. If2 < y1 < 5, then subsequent routings converge do the unique equilibrium routing y*, provided
k*
= r(s)ds i r ( s ) c / s for t > y c s
+
Jit, + JT:~~
+
= =
1 1 - $(R(y - E) 1 - R(1)
+
+ R(Y +
1 $(R(Y - E)
+ E.
E))
+ R(Y + 6))
with the understanding that R(-E) = R(0) = 0 and R ( l E ) = R(1). As before, given the current routing y, the new routing $ is the solution to (5). With load sharing, conclusions similar to those in the previous section can be drawn with less stringent stability conditions. They are summarized in the following propositions, which include the previous ones as special cases ( E = 0).
+
P r o p o s i t i o n 8 There exists a unique equilibrium routing under load sharing with any parameter E 2 0. Denote E O := ~ R ( Eand ) $r(s)ds.
€1
:= !j[R(l) - R ( l - E ) ] =
P r o p o s i t i o n 9 Given any routing y E (0, l ) , the new routing y given b y ( 5 ) exists and is unique if and only if k s / k ; is less o r equal to the minimum of ( 1 - EON1 - R(1) R,(1) - 2Eo
+
€0)
(1 - €1)(1- R(1) R( 1) - 2E1
+ €1)
It can be verified that the condition in Proposition 3 implies that i n the above proposition.
615
Proposition 10 Given any iniiial rouiing y1, the successive routings converge to the equilibrium routing y*, provided
-
-* ki r(t)
2
+
&J
where t := arg mintE(o,l)r ( t ) .
6
Conclusion
IGRP is widely used in practice. We have performed an exact analysis of its dynamic behavior on a simple ring network. It provides insight in setting the protocol parameters k,, k i . The distance metric in IGRP is the sum of traffic-sensitive and trafficinsensitive delay components, weighted by k,, ki. We have related the optimality (Theorem 2) and the stability (Corollaries 6 and 7) of the protocol to the ratio of these parameters. Roughly, the routing will converge t o the unique equilibrium routing if the trafficinsensitive component is given sufficient weight; otherwise, it will oscillate between two worst cases aft,er finitely many update periods.
References [I] D. Bertsekas and R. Gallager. Prentice-Hall Inc., 1987.
Data Networks.
[2] Dimitri P. Bertsekas. Dynamic behavior of shortest path routing algorithms for communication networks. IEEE Transactions on Automatic Control, pages 60-74, February 1982.
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